\(\tilde{L}_{b, N, T}\)  is U-statistic of order \(b\)  with kernel bounded between 0 and 1. By theorem A on p. 201 in [1]} it holds that \(\tilde{L}_{b, N, T}(x) - J_{b, T}(x)\xrightarrow{} 0\) .
Then, as in [2]}, for any \(\epsilon >0\)  with probability approaching it holds that
\(J(x-\epsilon )-\epsilon \le L_{b, N, T}\le J(x+\epsilon ) +\epsilon .\) 
